This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309528 #28 Nov 05 2019 06:00:08
%S A309528 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4,3,0,0,0,0,10,15,12,1,0,0,0,20,45,
%T A309528 72,37,2,0,0,0,35,105,252,266,117,6,0,0,0,56,210,672,1120,1044,333,14,
%U A309528 0,0,0,84,378,1512,3515,5270,3788,975,30,0,0,0,120,630,3024,9121,19350,23475,14056,2712,62,0
%N A309528 The number of non-equivalent distinguishing colorings of the cycle on n vertices with at most k colors (k>=1). The cycle graph is defined for n>=3; extended to n=1,2 using the closed form. Square array read by descending antidiagonals: the rows are indexed by n, the number of vertices of the cycle and the columns are indexed by k, the number of permissible colors.
%C A309528 A vertex-coloring of a graph G is called distinguishing if it is only preserved by the identity automorphism of G. This notion is considered in the subject of symmetry breaking of simple (finite or infinite) graphs. Two vertex-colorings of a graph are called equivalent if there is an automorphism of the graph which preserves the colors of the vertices. Given a graph G, we use the notation Phi_k(G) to denote the number of non-equivalent distinguishing colorings of G with at most k colors. The sequence here, displays A(n,k)=Phi_k(C_n), i.e., the number of non-equivalent distinguishing colorings of the cycle C_n on n vertices with at most k colors.
%H A309528 B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, <a href="https://arxiv.org/abs/1910.12102">Number of Distinguishing Colorings and Partitions</a>, arXiv:1910.12102 [math.CO], 2019.
%F A309528 A(n,k) = (A074650(n,k) - A284856(n,k))/2. - _Andrew Howroyd_, Aug 11 2019
%e A309528 The table begins:
%e A309528 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A309528 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A309528 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, ...
%e A309528 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, ...
%e A309528 0, 0, 12, 72, 252, 672, 1512, 3024, 5544, 9504, ...
%e A309528 0, 1, 37, 266, 1120, 3515, 9121, 20692, 42456, 80565, ...
%e A309528 0, 2, 117, 1044, 5270, 19350, 57627, 147752, 338364, 709290, ...
%e A309528 0, 6, 333, 3788, 23475, 102690, 355446, 1039248, 2673810, 6222150, ...
%e A309528 0, 14, 975, 14056, 106950, 555990, 2233469, 7440160, 21493836, 55505550, ...
%e A309528 0, 30, 2712, 51132, 483504, 3009426, 14089488, 53611992, 174189024, 499720518, ...
%e A309528 ------
%e A309528 For n=4, we can color the vertices of the cycle C_4 with at most 3 colors, in 3 ways, such that all the colorings distinguish the graph (i.e., no non-identity automorphism of C_4 preserves the coloring) and that all the three colorings are non-equivalent. The color classes are as follows:
%e A309528 { { 1 }, { 2 }, { 3, 4 } }
%e A309528 { { 1 }, { 2, 3 }, { 4 } }
%e A309528 { { 1, 2 }, { 3 }, { 4 } }
%o A309528 (PARI) A(n,k)={sumdiv(n, d, moebius(n/d)*(k^d/n - if(d%2, k^((d+1)/2), (k+1)*k^(d/2)/2)))/2} \\ _Andrew Howroyd_, Aug 11 2019
%Y A309528 Columns k=2..5 for n >= 3 are A032239, A032240, A032241, A032242.
%Y A309528 Cf. A074650, A284856.
%Y A309528 Different from A293496.
%K A309528 nonn,tabl
%O A309528 1,18
%A A309528 _Bahman Ahmadi_, Aug 06 2019